Representation of subharmonic functions in a~half-plane

K. G. Malyutin ; N. Sadik

Sbornik. Mathematics, Tome 198 (2007) no. 12, pp. 1747-1761

Résumé

The theory of subharmonic functions of finite order is based to a considerable extent on integral formulae. In the present paper representations are obtained for subharmonic functions in the upper half-plane with more general growth $\gamma(r)$ than finite order. The main result can be stated as follows. Let $\gamma(r)$ be a growth function such that either $\ln\gamma(r)$ is a convex function of $\ln r$ or the lower order of $\gamma(r)$ is infinite. Then for each proper subharmonic function $v$ of growth $\gamma(r)$ there exist an unbounded set $\mathbf R$ of positive numbers and a family $\{u_R:R\in\mathbf R\}$ of proper subharmonic functions in the upper half-plane $\mathbb{C}_+$ such that 1) the full measures of the $u_R$ in the discs $|z|\leqslant R$ are equal to the full measure of the function $v$; 2) $v-u_R\rightrightarrows0$ uniformly on compact subsets of $\mathbb{C}_+$ as $R\to\infty$, $R\in\mathbf R$; 3) the function family $\{u_R:R\in\mathbf R\}$ satisfies the growth constraints uniformly in $R$, that is, $T(r,u_R)\leqslant A\gamma(Br)/r$, where $A$ and $B$ are constants and $T(r,\,\cdot\,)$ is the growth characteristic. Bibliography: 16 titles.